Difficulty: Introductory | Prerequisites: Basic algebra and trigonometry
Big picture: Motion and vectors form the foundation of classical mechanics. This material sits right at the start of most physics courses because everything that follows, from Newton's laws to energy and momentum, builds on your ability to describe how objects move and to work with quantities that have both size and direction. If you are coming in cold, you need comfort with basic algebra and right-triangle trigonometry. If you have those, you are ready.
Motion describes how an object's position changes over time, and the three quantities you need to pin it down are displacement, velocity and acceleration. All three are vectors, meaning they carry both a magnitude and a direction. Master how to add, subtract and resolve vectors and you have the toolkit for everything else in kinematics.
Motion
A change in the position of an object over time. Described using displacement, velocity and acceleration.
In simple terms, any time something moves from one place to another, that is motion.
Displacement
A vector quantity equal to the final position minus the initial position of an object. It tells you how far and in what direction the object ended up from where it started.
Think of it as the straight-line shortcut from start to finish, regardless of the actual path taken.
Velocity
A vector quantity that describes the rate of change of displacement. Equal to displacement divided by the time interval.
In simple terms, velocity is speed with a direction attached. 60 km/h north is a velocity; 60 km/h on its own is just a speed.
Average velocity
The total displacement divided by the total time interval. Useful when velocity is not constant throughout the motion.
Acceleration
A vector quantity that describes the rate of change of velocity. Equal to the change in velocity divided by the time interval.
Think of it as how quickly your velocity is changing. Speeding up, slowing down, or turning all count as acceleration.
Vector
A quantity that has both magnitude (size) and direction. Displacement, velocity and acceleration are all vectors. Contrast with a scalar, which has magnitude only (e.g. temperature, mass).
Scalar
A quantity with magnitude but no direction. Speed, distance and mass are scalars. Often confused with their vector counterparts.
Motion is always measured relative to a reference point (origin). Without a coordinate system, "position" has no meaning.
Displacement = final position minus initial position. It is a vector: direction matters.
An object that walks 3 m east then 3 m west has zero displacement but 6 m of distance travelled.
Velocity = displacement / time. It is also a vector.
Average velocity covers an entire interval: total displacement divided by total time.
Instantaneous velocity is the velocity at a single moment (the limit as the time interval shrinks to zero).
Acceleration = change in velocity / time. Again, a vector.
An object can accelerate even at constant speed if its direction changes (e.g. circular motion).
Along a straight line, vectors reduce to signed numbers: positive for one direction, negative for the other.
The kinematic equation for constant-acceleration 1-D motion is: d = v₀t + (1/2)at²
d = displacement
v₀ = initial velocity
a = acceleration (constant)
t = time elapsed
This equation assumes acceleration does not change during the interval.
In 2-D, displacement is a vector in the x-y plane.
Each component (x and y) can be treated independently.
A projectile's horizontal velocity and vertical acceleration are handled as separate 1-D problems.
The magnitude of the displacement vector is found using the Pythagorean theorem: |d| = √(dx² + dy²)
The direction is found using trigonometry: θ = arctan(dy / dx)
Addition: Place vectors tip-to-tail. The resultant runs from the first tail to the last tip. Algebraically, add corresponding components: Rₓ = Aₓ + Bₓ, Rᵧ = Aᵧ + Bᵧ.
Subtraction: Reverse the direction of the vector being subtracted, then add.
Magnitude: Use the Pythagorean theorem on the components.
Direction: Use inverse trigonometric functions (typically arctan) on the component ratio.
Kinematic equation (constant acceleration, 1-D)
d = v₀t + ½at²
where d = displacement, v₀ = initial velocity, a = constant acceleration, t = time.
Vector magnitude (2-D)
|v| = √(vₓ² + vᵧ²)
Vector direction (2-D)
θ = arctan(vᵧ / vₓ)
Always check which quadrant the vector sits in; arctan alone gives values in the first and fourth quadrants only.
Vector addition (component method)
Rₓ = Aₓ + Bₓ
Rᵧ = Aᵧ + Bᵧ
Then find |R| and θ from the components above.
GPS navigation systems calculate your displacement vector in real time to determine how far you are from your destination and in which direction to travel. The device resolves velocity into components to update your position on the map.
Engineers designing car safety systems use acceleration data (from accelerometers) to detect a sudden change in velocity during a collision and trigger airbag deployment within milliseconds.
Students often confuse displacement with distance. Displacement is a vector (start to finish, straight line); distance is a scalar (total path length). An object that returns to its starting point has zero displacement but non-zero distance.
Speed and velocity are frequently used interchangeably, but speed is a scalar (no direction) and velocity is a vector. An object moving at constant speed in a circle has a changing velocity because its direction keeps changing.
Many students assume that zero velocity means zero acceleration. An object thrown straight up has zero velocity at the peak of its arc, but gravity still accelerates it at 9.8 m/s² downward at that instant.
When adding vectors, students sometimes add magnitudes directly. This only works if the vectors point in the same direction. In all other cases, you must use the component method or the tip-to-tail geometric approach.
⚠️ You will almost certainly be asked to distinguish between distance and displacement, and between speed and velocity. Know the vector/scalar distinction cold.
⚠️ The kinematic equation d = v₀t + ½at² appears in nearly every introductory physics exam. Be comfortable rearranging it to solve for any of the four variables.
⚠️ Vector component problems (resolving a vector into x and y, then recombining) are a staple of exam questions. Practise until the process feels mechanical.
⚠️ Watch for problems where acceleration is zero but velocity is not, or where velocity is zero but acceleration is not. These test whether you understand each quantity independently.
True or false: Displacement and distance are always equal. (False. They are equal only when the object moves in a straight line without reversing direction.)
Fill in the blank: Velocity is a ____ quantity, while speed is a ____ quantity. (vector; scalar)
True or false: An object with zero velocity must have zero acceleration. (False. Think of a ball at the top of its trajectory: velocity is zero, but acceleration due to gravity is still 9.8 m/s² downward.)
Fill in the blank: To find the magnitude of a 2-D vector from its components, use the ____ theorem. (Pythagorean)
True or false: You can add the magnitudes of two vectors to get the magnitude of the resultant. (False, unless the vectors point in exactly the same direction.)
Q: A car drives 5 km north, then 3 km south. What is its displacement? What is the total distance travelled?
A: Displacement is 2 km north (5 km minus 3 km in the same line). Total distance travelled is 8 km.
Q: An object starts from rest and accelerates at 4 m/s² for 3 seconds. What is its displacement?
A: Using d = v₀t + ½at², with v₀ = 0, a = 4 m/s², t = 3 s: d = 0 + ½(4)(9) = 18 m.
Q: A vector has components Aₓ = 3 and Aᵧ = 4. What are its magnitude and direction?
A: Magnitude = √(3² + 4²) = √25 = 5. Direction = arctan(4/3) ≈ 53.1° above the positive x-axis.
Q: Why is velocity a vector while speed is a scalar?
A: Velocity includes directional information (e.g. 20 m/s east), while speed is only the magnitude of velocity (20 m/s) with no direction specified.
Q: Can an object have constant speed but changing velocity? Give an example.
A: Yes. An object moving in a circle at constant speed has a continuously changing velocity because its direction of travel changes at every point along the circle.
This material connects directly to Newton's laws of motion, which build on the concepts of velocity and acceleration by introducing force as the cause of acceleration (F = ma). Without a solid grasp of vectors and kinematics, Newton's second law is difficult to apply in two dimensions.
Projectile motion, circular motion and relative motion all extend the vector framework introduced here into more complex scenarios. Each of those topics assumes you can resolve vectors into components and apply the kinematic equations independently along each axis.
Energy and momentum, covered later in most physics courses, also depend on velocity as a vector. Kinetic energy uses speed (the magnitude), while momentum is a vector quantity that carries the direction of velocity forward.
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